Secants, bitangents, and their congruences

0.64 MB·

by  Kathlén Kohn

Secants, bitangents, and their congruences
Kathle´nKohn,BerntIvarUtstølNødland,andPaoloTripoli
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Abstract AcongruenceisasurfaceintheGrassmannianGr(1,P3)oflinesinpro-
3
jective3-space.ToaspacecurveC,weassociatetheChowhypersurfaceinGr(1,P3)
1
consistingofalllineswhichintersectC.Wecomputethesingularlocusofthishy-
] persurface, which contains the congruence of all secants to C. A surface S in P3
G defines the Hurwitz hypersurface in Gr(1,P3) of all lines which are tangent to S.
A WeshowthatitssingularlocushastwocomponentsforgeneralenoughS:thecon-
. gruence of bitangents and the congruence of inflectional tangents. We give new
h
t proofs for the bidegrees of the secant, bitangent and inflectional congruences, us-
a
inggeometrictechniquessuchasduality,polarlociandprojections.Wealsostudy
m
the singularities of these congruences. In order to make this article self-contained
[ andaccessibletoabroaderaudience,wediscusssomeoftherelevantbackground

1 materialindetail.
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7 1 Introduction
3
0
. The aim of this article is to study subvarieties of Grassmannians which naturally
1
arise from subvarieties of complex projective 3-space P3 :=P3. In particular we
0 C
7 aremostlyinterestedinthreefoldsandsurfacesinGr(1,P3),classicallycalledline
1 complexes and congruences, respectively. We are interested in determining their
:
v
i
X Kathle´nKohn
InstituteofMathematics,TUBerlin,SekretariatMA6-2,Straßedes17.Juni136,10623Berlin,
r Germany,e-mail:[email protected]
a
BerntIvarUtstølNødland
Department of Mathematics, University of Oslo, Moltke Moes vei 35, Niels Henrik Abels hus,
0851Oslo,Norway,e-mail:[email protected]
PaoloTripoli
Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7AL, United
Kingdom,e-mail:[email protected]
1
2 Kathle´nKohn,BerntIvarUtstølNødland,andPaoloTripoli
classes in the Chow ring of Gr(1,P3) and their singular loci. In the following, the
singular points of a congruence are simply its singular points as a subvariety of
the Grassmannian Gr(1,P3), although classically a singular point of a congruence
is a point in P3 through which there pass infinitely many lines of the congruence.
Nowadays,thelatterpointsarecalledfundamentalpoints.
The Chow complex CH (C) of a curve C ⊂ P3 is the variety of all lines in
0
Gr(1,P3) intersectingC. The Hurwitz complex CH (S) of a surface S in P3 is the
1
closureofthesetofalllinesinGr(1,P3)whicharetangenttoSatasmoothpoint.
Ourmainresultsarethefollowing:
Theorem1.1.LetC beaspacecurvethatisnotcontainedinaplaneandhasde-
gree d, geometric genus g, and ordinary singularities x ,...,x with multiplicities
1 s
r ,...,r .Then
1 s
• Sing(CH (C))=Sec(C)∪(cid:83)s (cid:8)L∈Gr(1,P3)|x ∈L(cid:9),whereSec(C)isthelo-
0 i=1 i
cusofsecantlinestoC(seeTheorem3.7).
• ThebidegreeofSec(C)is
(cid:32) (cid:33)
1 s 1 1
(d−1)(d−2)−g−∑ r(r −1), d(d−1)
i i
2 2 2
i=1
(seeTheorem3.9).
• If C is smooth, then the singular locus of Sec(C) consists of all lines that in-
tersectC with total multiplicity of at least three (see Propositions 7.1, 7.2 and
Theorem7.4).
LetS⊂P3beageneralsurfaceofdegreed≥4.Then
• Sing(CH (S))=Bit(S)∪Infl(S),whereBit(S)isthelocusofbitangentstoSand
1
Infl(S)isthelocusofinflectionaltangentstoS(seeTheorem4.2).
• ThebidegreeofBit(S)is
(cid:18) (cid:19)
1 1
d(d−1)(d−2)(d−3), d(d−2)(d−3)(d+3)
2 2
(seeTheorem4.3).
• ThebidegreeofInfl(S)is
(d(d−1)(d−2), 3d(d−2))
(seeTheorem4.3).
• ThesingularlocusofInfl(S)consistsofalllinesthatareinflectionaltangentsat
atleasttwopointsofSorthatintersectSwithmultiplicityatleastfouratsome
point(seeTheorem7.5).
ThebidegreeofInfl(S)wasalreadycalculatedin [23,Prop.4.1].Thebidegrees
ofBit(S),Infl(S)andSec(C)(onlyforsmoothC)werealreadyprovenin[2],apaper
towhichweoweagreatdebt.However,wegivealternative,moregeometricproofs,
Secants,bitangents,andtheircongruences 3
notrelyingonChernclasstechniques.ThesingularlocusofSec(C)wasalsopartly
describedin[2,Lemma2.3].
Usingdualitywecanalsoshowrelationsbetweensomeoftheabovevarieties.
Theorem1.2.IfC is a smooth space curve that is not contained in a plane, then
thesecantlinesofC aredualtothebitangentlinesofthedualsurfaceC∨ andthe
tangentlinesofCaredualtotheinflectionaltangentlinesofC∨(seeTheorem5.5).
Congruences and line complexes have been actively studied both in the nine-
teenthcenturyandinmoderntimes.ThestudyofcongruencesgoesbacktoKum-
mer [21], who classified those of order one, where the order of a congruence is
thenumberoflinesinthecongruencethatpassthroughageneralpointin3-space.
Specifically Chow complexes of space curves were first introduced by Cayley [4]
andlatergeneralizedtoarbitraryvarietiesbyChowandvanderWaerden[5].Many
classical results from the second half of the 19th century can be found in Jes-
sop’s monograph [17]. Hurwitz complexes and further generalizations to general
Grassmanniansknownashigherassociated orcoisotropichypersurfacesarestud-
ied in[11, 19, 30]. Catanese [3] showed that the Chow complexes of space curves
andHurwitzcomplexesofsurfacesareexactlytheself-dualcomplexesintheGrass-
mannianGr(1,P3).Ran[26]studiedsurfacesoforderoneingeneralGrassmannians
andgaveamodernproofofKummer’sclassification.Congruencesplayaroleinal-
gebraicvision/multi-viewgeometry,wherecamerasaremodeledasmapsfromP3
to congruences [25]. The multidegree of the image of several of those cameras is
computedinthearticle[9]byEscobarandKnutsoninthisvolume.
In Section 2 we recall preliminary facts about Grassmannians and their subva-
rieties. In Section 3 we study the singular locus of the Chow complex of a space
curveandwecomputeitsbidegree.Section4considersHurwitzcomplexesaswell
as bitangent and inflectional congruences. We give a new proof for the bidegrees
ofthesecongruencesusingonlygeometricarguments.Ourproofusestheconcept
of projective duality, which we introduce in Section 5. For expository reasons we
giveanintroductiontotheintersectiontheoryofGr(1,P3),Chernclassesandtheir
relations to the questions studied in this paper in Section 6. This is because most
modern formulations of questions of this sort are written and studied using these
techniques. We consider some example questions that can be easily computed us-
ing intersection theory, for example we compute the number of lines bitangent to
twogeneralsurfacesofgivendegree.Atlast,westudythesingularlociofsecant,
bitangentandinflectionalcongruencesinSection7,andwedescribecomputational
aspectsinSection8.
This project was inspired by some of the problems in [31], in particular Prob-
lem 5 on Curves, Problem 4 on Surfaces and Problem 3 on Grassmannians. We
give,throughoutthearticle,fullanswerstotheseproblems.
4 Kathle´nKohn,BerntIvarUtstølNødland,andPaoloTripoli
2 PreliminariesonGrassmannians
TheGrassmannianGr(1,P3)oflinesinP3isafour-dimensionalvariety,whichcan
be embbedded into P5 via the Plu¨cker embedding. A line that is spanned by two
distinct points (x :x :x :x ),(y :y :y :y )∈P3 is uniquely identified with
0 1 2 3 0 1 2 3
(p : p : p : p : p : p ),where p istheminorformedofcolumnsiand j
01 02 03 12 13 23 ij
of the matrix (x0 x1 x2 x3). The entries p computed in this way are called Plu¨cker
y0 y1 y2 y3 ij
coordinatesandsatisfythePlu¨ckerrelationp p −p p +p p =0.Allpoints
01 23 02 13 03 12
inP5satisfyingthisrelationarethePlu¨ckercoordinatesofsomeline.Analogously,
one can also write lines as the intersection of two planes {x∈P3 |a x +a x +
0 0 1 1
a x +a x =0} and {x∈P3 |b x +b x +b x +b x =0}, and consider the
2 2 3 3 0 0 1 1 2 2 3 3
minorsq ofthematrix(cid:0)a0 a1 a2 a3(cid:1).TheseminorssatisfythesamePlu¨ckerrelation
ij b0 b1 b2 b3
andarecalleddualPlu¨ckercoordinates.Wecangofromoneconventiontotheother
viathemap
p (cid:55)→q , p (cid:55)→−q , p (cid:55)→q , p (cid:55)→q , p (cid:55)→−q , p (cid:55)→q .
01 23 02 13 03 12 12 03 13 02 23 01
WestudysubvarietiesofthisGrassmannianGr(1,P3).Anintroductiontosubva-
rieties of general Grassmannians is given in [1]. In particular, we are interested in
theirsingularlociandincomputingtheirdegree.Thereisanaturalnotionofdegree
forasubvarietyΣ⊂Gr(1,P3).Formallythiscanbedefinedasthecoefficientsofthe
classofΣ intheChowringofGr(1,P3)withrespecttothebasisgivenbySchubert
cycles.DetailsaboutthiscanbefoundinSection6.Fornow,wegivethefollowing,
moregeometric,definitions.
Threefolds in Gr(1,P3) are classically called line complexes. Let H ⊂P3 be a
general plane and let v∈H be a general point. The degree of a line complex Σ ⊂
Gr(1,P3)isdefinedasthenumberoflinesL∈Σ withv∈L⊂H.Anexampleofa
linecomplexistheChowcomplexCH (C)ofacurveC⊂P3.AgeneralplaneH
0
intersectsC in deg(C) many points, so that there are deg(C) many lines in H that
passthroughageneralpointvinH andintersectC (seeFig.1).Hence,thedegree
oftheChowcomplexisequaltothedegreeofthecurve.
C
v H
Fig.1 degCH (C)=degC.
0
1
Secants,bitangents,andtheircongruences 5
SurfacesinΣ ⊂Gr(1,P3)areclassicallyknownascongruences.Thedegreeof
acongruenceisinfactabidegree(α,β).Theorderα isthenumberoflinesL∈Σ
thatpassthroughageneralpointv∈P3,whereastheclassβ isthenumberoflines
L∈Σ that lie in a general plane H ⊂P3. Consider for example the congruence
of lines passing through a fixed point x. Given a general point p, this congruence
containsauniquelinepassingthoughp(thelinespannedbyxandp).Givenaplane
H,wehaveingeneralx∈/H,andthecongruencedoesnotcontainanylinecontained
inH.Thisshowsthatthesetoflinespassingthroughafixedpointisacongruence
withbidegree(1,0).Analogously,thecongruenceoflineslyinginafixedplanehas
bidegree(0,1).
The degree of a curve Σ ⊂Gr(1,P3) is the number of lines L∈Σ intersecting
agenerallineofP3.EquivalentlyitisthenumberoflinesintheintersectionofΣ
withtheChowcomplexofageneralline.AnexampleofacurveinGr(1,P3)isthe
setofalllinesthatlieinafixedplaneH⊂P3andpassthroughafixedpointv∈H.
ThiscurveisinfactalineinGr(1,P3).Weseedirectlythatitsdegreeisonesincea
generallinehasoneintersectionpointwithH andthereisexactlyonelinepassing
throughthisintersectionpointandv.
Finally,thedegreeofazero-dimensionalsubvarietyΣ ⊂Gr(1,P3)issimplythe
numberoflinesL∈Σ.
3 Secants
InthissectionwecomputethesingularlocusoftheChowcomplexofaspacecurve,
andwegiveaformulaforthebidegreeofthesecantcongruenceincasethecurve
hasmildsingularities.
A curveC⊂P3 is defined by two or more homogeneous polynomials, and the
polynomials used to describe the curve are not unique. There is however a way
to describeC with a single polynomial equation. For this, we consider the Chow
complex
CH (C):={L|C∩L(cid:54)=0/}⊂Gr(1,P3).
0
Thisisalinecomplex,i.e.,ahypersurfaceinGr(1,P3).Itisdefinedbyonepolyno-
mialinthePlu¨ckercoordinatesofGr(1,P3),whichisuniqueuptoscalingandthe
Plu¨ckerrelation.ThispolynomialisknownastheChowformofC.Aniceintroduc-
tiontoChowformsisgivenin[6].
Example3.1([6, Sec. 1.2, Prop. 1.2]). The twisted cubic is parametrized by (s3 :
s2t :st2 :t3). The line L given by the intersection of the planes {x∈P3 |a x +
0 0
a x +a x +a x =0}and{x∈P3|b x +b x +b x +b x =0}intersectsthe
1 1 2 2 3 3 0 0 1 1 2 2 3 3
twistedcubicifandonlyifthereexists(s:t)∈P1suchthat
a s3+a s2t+a st2+a t3=0=b s3+b s2t+b st2+b t3. (1)
0 1 2 3 0 1 2 3
6 Kathle´nKohn,BerntIvarUtstølNødland,andPaoloTripoli
Hence,weneedtocomputetheresultantofthetwocubicpolynomialsin(1),i.e.,the
polynomialinthecoefficientsa,b thatvanishesexactlywhenthetwopolynomials
i j
haveacommonroot.Inthiscase,thisisthedeterminantoftheSylvestermatrix
 
a a a a 0 0
3 2 1 0
 0 a3 a2 a1 a0 0 
 
 0 0 a3 a2 a1 a0.
b3 b2 b1 b0 0 0 
 
 0 b3 b2 b1 b0 0 
0 0 b b b b
3 2 1 0
This determinant can be expressed in the dual Plu¨cker coordinates q that are the
ij
minorsofthematrix(cid:0)a0 a1 a2 a3(cid:1).Itisequaltothedeterminantofthe3×3-matrix
b0 b1 b2 b3
 
q q q
01 02 03
q02 q03+q12 q13,
q q q
03 13 23
whichistheChowformofthetwistedcubic.
WheneverC has degree at least two, the set of lines that intersect the curveC
attwopointsisacongruence,i.e.,asurfaceinGr(1,P3).Thisiscalledthesecant
congruenceofC.Morepreciselyitisdefinedas
Sec(C):={L|∃x,y∈Reg(C):x,y∈L, x(cid:54)=y}⊂Gr(1,P3),
whereReg(X)denotesthesetofsmoothpointsofaprojectivevarietyX.Westress
thatalineintersectingthecurveonlyatasingularpointmightnotbecontainedin
thesecantcongruence,althoughithasintersectionmultiplicityatleasttwowiththe
curve at that point (see Remark 3.8). For smooth curves of degree at least two the
secantcongruenceisthesingularlocusoftheChowcomplex.Forsingularcurves,
thesingularlocusoftheChowcomplexcontainsoneadditionalcomponentforeach
singular point, namely all the lines passing through the point. Our key ingredient
to prove this is Proposition 3.2. Note that, since in this paper we only work with
varietiesembeddedinaprojectivespace,bytangentspaceT X ofavarietyX ⊂Pn
x
atapointx∈X wewillalwaysmeantheembeddedtangentspace
(cid:40) (cid:12) (cid:41)
(cid:12) n ∂f
T X := y∈Pn(cid:12)∀f ∈I(X):∑ (x)·y =0 .
x (cid:12) ∂X i
(cid:12) i=0 i
Proposition3.2.LetX,Y beirreducibleprojectivevarietieswithabirational,finite
and surjective morphism f :X →Y, and let y∈Y. ThenY is smooth at y if and
onlyifthefiber f−1(y)containsexactlyonepointx∈X,X issmoothatxandthe
differentiald f :T X →TY isaninjection.
x x y
WewillprovethiswiththehelpofZariski’sMainandConnectednessTheorems
aswellascriteriaforfinitemapsandisomorphisms.
Secants,bitangents,andtheircongruences 7
Theorem3.3(Zariski’s Main Theorem). Let Y be a normal irreducible variety
and let f :X →Y be a birational morphism with finite fibers from an irreducible
varietyX toY.Then f isanisomorphismofX withanopensubsetofY.
Theorem3.4(Zariski’sConnectednessTheorem).LetX,Y beirreducibleprojec-
tivevarietieswithabirationalmorphism f :X →Y,andlety∈Y.IfY isnormalat
y,then f−1(y)isaconnectedsetintheZariskitopology.
Lemma3.5(Criterion for Finite Maps). Let X,Y be irreducible projective vari-
etieswithamorphism f :X →Y.LetY ⊂Y beopen,X := f−1(Y ),andlet f be
0 0 0 0
therestrictionof f toX .Then f isfiniteifandonlyifitsfibersarefinite.
0 0
Theorem3.6(CriterionforIsomorphism).LetX,Y beirreduciblevarietieswith
afinitemap f :X →Y.Then f isanisomorphismifandonlyifitisbijectiveand
andthedifferentiald f :T X →T Y isaninjectionforallx∈X.
x x f(y)
Theaboveformulationof Zariski’sMain andConnectedness Theoremscan for
examplebefoundin[22,Ch.III.9,pp.288–289,Thm.I,Thm.V],whereasthetwo
criteriaappearin[14,Lem.14.8,Thm14.9].
Proof (Proposition 3.2). First assume thatY is smooth at y. Then it follows from
Theorem3.4thatthefiber f−1(y)isconnected.Since f isfinite,ithasfinitefibers
and f−1(y)={x}forsomex∈X.LetY betheopensetofsmoothpointsinY,and
0
letX := f−1(Y ).Thentherestrictionof f toX isanisomorphismofX withY
0 0 0 0 0
byTheorem3.3.Inparticular,wehavethatx∈X ⊂X isasmoothpoint.Moreover,
0
Theorem3.6yieldsthatthedifferentiald f isaninjection.
x
Fortheotherdirection,weassumethat f−1(y)={x}forasmoothpointx∈X
withinjectivedifferentiald f.LetY beanopenneighborhoodofycontainingonly
x 1
pointsinY withone-elementfibersandinjectivecorrespondingdifferentials.Then,
byTheorem3.6andLemma3.5,wegetanisomorphismofX := f−1(Y )withY .
1 1 1
Sincex∈X issmooth,wealsohavethaty∈Y ⊂Y issmooth. (cid:116)(cid:117)
1 1
Havingthis,wecandescribethesingularlocusoftheChowcomplexofagiven
spacecurve.
Theorem3.7.LetC⊂P3 be an irreducible curve of degree at least two. Then we
have
Sing(CH (C))=Sec(C)∪ (cid:91) (cid:8)L∈Gr(1,P3)|x∈L(cid:9).
0
x∈Sing(C)
Proof. Wewillfirstshowthattheincidencevariety
A :={(x,L)|x∈L}⊂C×Gr(1,P3)
C
issmoothat(x,L)ifandonlyifx∈Cissmooth.Forthis,let f ,...,f begenerators
1 k
for the vanishing ideal of C, and consider the affine chart where x (cid:54)=0 and the
0
Plu¨ckercoordinatep (cid:54)=0.Thenwemayassumex=(1:α:β :γ)andLisspanned
01
by(1:0:a:b)and(0:1:c:d).Wehavethatx∈LifandonlyifListherowspace
of
8 Kathle´nKohn,BerntIvarUtstølNødland,andPaoloTripoli
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
1α β γ 1α 10β−αcγ−αd
= ,
0 1 c d 0 1 01 c d
which is equivalent to a=β−αc and b=γ−αd. Hence, in the described affine
chart,A canbewrittenas
C
{(α,β,γ,a,b,c,d)| f(1,α,β,γ)=0fori=1,...,k, a=β−αc, b=γ−αd}.
i
(2)
As A has dimension three, it is smooth at (x,L) if and only if its tangent space,
C
thatisthekerneloftheJacobianmatrix,hasdimensionthreeor,equivalently,ifthe
Jacobianmatrixhasrankfour.Thismatrixhastheform
∂f1(1,α,β,γ) ∂f1(1,α,β,γ) ∂f1(1,α,β,γ) 0 0 0 0 
∂α ∂β ∂γ
 . . . 
 .. .. .. 
 
JAC :=∂fk(1,α,β,γ) ∂fk(1,α,β,γ) ∂fk(1,α,β,γ) 0 0 0 0 .
 ∂α ∂β ∂γ 
 −c 1 0 −1 0 −α 0 
−d 0 1 0 −1 0 −α
ThereforeithasrankfourifandonlyiftheJacobianmatrixofChasranktwoor,in
otherwords,ifandonlyifx∈C issmooth.ThisshowsthatA issmoothat(x,L)
C
exactlywhenxissmoothinC.
ByLemma3.5,theprojection
π :A −→CH (C),
C 0
(3)
(x,L)(cid:55)−→L
is finite, since otherwise C would contain a line, contradicting our assumptions.
Moreover,thegeneralfiberofπ hascardinalityoneasthegenerallineL∈CH (C)
0
intersectsCinasinglepoint.Hence,π isbirational.ApplyingProposition3.2onπ
showsthatCH (C)issmoothatLifandonlyifπ−1(L)={(x,L)}wherex∈C is
0
asmoothpointandthedifferentiald π isinjective.Usingthesameaffinechart
(x,L)
as above, we have that the differential d π projects every element in the kernel
(x,L)
ofJ onitslastfourcoordinates.Thismapisnotinjectiveifandonlyifthekernel
AC
ofJ containsanelementoftheform(∗,∗,∗,0,0,0,0)T (cid:54)=0.Suchanelementisin
AC
thekernelofJ ifandonlyifitisequalto(τ,cτ,dτ,0,0,0,0)T forτ∈C\{0}and
AC
(cid:18) (cid:19)
∂f ∂f ∂f
i i i
+c +d (1,α,β,γ)=0fori=1,...,k.
∂α ∂β ∂γ
Hence,forasmoothpointx∈C,thedifferentiald π isnotinjectiveifandonly
(x,L)
ifListhetangentlineofCatx.
Sincewehavethat|π−1(L)|=1ifandonlyifLisnotasecantlineandsinceall
tangentlinestoCarecontainedinSec(C),itfollowsthatCH (C)issmoothatLif
0
andonlyifL∈/Sec(C)andLmeetsCatasmoothpoint. (cid:116)(cid:117)
Secants,bitangents,andtheircongruences 9
Remark3.8.We can see with local computations that the secant congruence ofC
generallydoesnotcontainalllinesthroughsingularpointsofC.Forthis,letx∈C
beanordinarysingularity,i.e.,xistheintersectionofr≥2branchesofC andthe
rtangentlinesofthebranchesarepairwisedifferent.Wewillnowshowthataline
L intersecting C only at x is contained in Sec(C) if and only if L lies in a plane
spannedbytwoofther tangentlinesatx.Theunionofallthoselinesformtheso
called tangent star ofC at x. This notion has been introduced in [18] and studied
in[28].
Assumex=(1:0:0:0)andL∈Sec(C)intersectsConlyatx.ThelineLmust
be the limit of lines L that intersectC at two distinct smooth points. Without loss
t
ofgenerality,L isnotoneofthetangentlinesofC atx andeachlineL intersects
t
Catatleasttwodistinctbranches.Sincethereareonlyfinitelymanybranches,we
canfurtherassumethatthelinesL intersectthesametwobranchesofC.Thesetwo
t
branchesareparametrizedby(1: f (t): f (t): f (t))and(1:g (t):g (t):g (t)),
1 2 3 1 2 3
respectively,with f(0)=0=g (0).Thenthetwotangentlinesatthesebranchesare
i j
spannedbyxand(1: f(cid:48)(0): f(cid:48)(0): f(cid:48)(0))or(1:g(cid:48)(0):g(cid:48)(0):g(cid:48)(0)),respectively.
1 2 3 1 2 3
ThelinesL intersectthefirstbranchat(1: f (ϕ(t)): f (ϕ(t)): f (ϕ(t)))andthe
t 1 2 3
secondbranchat(1:g (ψ(t)):g (ψ(t)):g (ψ(t))),whereϕ(0)=0=ψ(0).The
1 2 3
Plu¨ckercoordinatesofL are
t
(cid:18)
g (ψ(t))−f (ϕ(t)) g (ψ(t))−f (ϕ(t)) g (ψ(t))−f (ϕ(t))
1 1 2 2 3 3
: : :
t t t
(cid:19)
f (ϕ(t))g (ψ(t))−f (ϕ(t))g (ψ(t))
1 2 2 1 :... .
t
Lettingt gotozero,wegetthelineLwithPlu¨ckercoordinates
(cid:0)g(cid:48)(0)ψ(cid:48)(0)−f(cid:48)(0)ϕ(cid:48)(0):g(cid:48)(0)ψ(cid:48)(0)−f(cid:48)(0)ϕ(cid:48)(0):g(cid:48)(0)ψ(cid:48)(0)−f(cid:48)(0)ϕ(cid:48)(0) :
1 1 2 2 3 3
0:0:0).
Thislineisspannedbyxand
(1:g(cid:48)(0)ψ(cid:48)(0)−f(cid:48)(0)ϕ(cid:48)(0):g(cid:48)(0)ψ(cid:48)(0)−f(cid:48)(0)ϕ(cid:48)(0):g(cid:48)(0)ψ(cid:48)(0)−f(cid:48)(0)ϕ(cid:48)(0)),
1 1 2 2 3 3
andthusliesintheplanespannedbythetwotangentlines.
Moreover,weseefromthiscomputationthatalllinesthatpassthroughxandlie
intheplanespannedbythetangentlinescanbeapproximatedbylinesthatintersect
both of the branches at points different from x. For this, one only needs to choose
ϕ(t)=λt andψ(t)=µt forallpossibleλ,µ ∈C\{0}.
The bidegree of the secant congruence of a smooth curve was calculated in [2,
Prop.2.1]withChernclasses.Herewegiveageometricdescriptionofthisbidegree
also for curves with ordinary singularities. As in Remark 3.8, an ordinary singu-
larity is the intersection of r≥2 branches of the curve and the r tangent lines of
the branches are pairwise different. The number r is called the multiplicity of the
ordinarysingularity.Ifr=2,thesingularityiscalleda(simple)node.
10 Kathle´nKohn,BerntIvarUtstølNødland,andPaoloTripoli
Theorem3.9.LetC⊂P3beanirreduciblecurvethatisnotcontainedinanyplane
andhasonlyordinarysingularitiesx ,...,x withmultiplicitiesr ,...,r .Letd de-
1 s 1 s
notethedegreeandgdenotethegeometricgenusofthecurve.Thenthebidegreeof
thesecantcongruenceSec(C)is(12(d−1)(d−2)−g−∑si=112ri(ri−1),21d(d−1)).
Proof. Let H ⊂P3 be a general plane. The intersection of H withC consists of d
points.AnytwoofthesepointsdefineasecantlinelyinginH (seeFig.2).Hence,
thereare(cid:0)d(cid:1)secantlinescontainedinH.ThisistheclassofSec(C).
2
C
H
Fig.2 class(Sec(C))= 1deg(C)(deg(C)−1).
2
1
Letv∈P3beageneralpoint.WeareinterestedincalculatingtheorderofSec(C),
which is the number of secant lines passing through v. The point v defines a pro-
jection π :P3 (cid:57)(cid:57)(cid:75)P2. LetC(cid:48) be the image ofC under π . A line passing through
v v
vandintersectingCattwopointsgetsmappedbyπ toasimplenodeoftheplane
v
curveC(cid:48) (see Fig. 5). Every ordinary singularity ofC gets mapped to an ordinary
singularityofC(cid:48)withthesamemultiplicity.TheplanecurveC(cid:48)hasthesamedegree
as the space curve C. Moreover, as the geometric genus is invariant under bira-
tional transformation (see [15, Thm. II.8.19]), it also has the same genus. By the
genus-degreeformula forplane curves[27, p.54, Eq.(7)], thegenus g isequal to
21(d−1)(d−2)−∑si=121ri(ri−1)minusthenumberofsecantsofCpassingthrough
v.Thisconcludestheproof. (cid:116)(cid:117)
Remark3.10.IfC⊂P3 is a curve of degree d ≥2 which is contained in a plane,
then its secant congruence consists of all lines in that plane and thus has bidegree
(0,1).
Problem 5 on Curves in [31] asks to compute the dimension and bidegree of
Sing(CH (C)).WealreadyknowbyTheorem3.7thatitsdimensionistwoifC is
0
notaline.Forcompletenesswealsostateitsbidegreeexplicitly.
Corollary3.11.LetC⊂P3 be an irreducible curve of degree d ≥2 that has only
ordinary singularities x ,...,x with multiplicities r ,...,r . Let g denote the geo-
1 s 1 s
metricgenusofthecurve.ThenthebidegreeofSing(CH (C))isequalto
0
(cid:32) (cid:33)
1 s 1 1
(d−1)(d−2)−g−∑ r(r −1)+s, d(d−1)
i i
2 2 2
i=1